It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled)
polyhedron based on a rectangle that meets every vertical line in a segment,
has a grid unfolding: its surface may be unfolded to a single non-overlapping
piece by cutting along grid edges defined by coordinate planes through every
vertex.Comment: 7 pages, 7 figures, 5 references. First revision adds Figure 7, and
  improves Figure 6. Second revision further improves Figure 7, and adds one
  clarifying sentence. Third corrects label in Figure 7. Fourth revision
  corrects a sentence in the conclusion about the class of shapes now known to
  be grid-unfoldabl

O'Rourke, Joseph

English

arXiv

It is shown that every orthogonal terrain, i.e., an orthogonal (rightangled) polyhedron based on a rectangle that meets every vertical line in a segment, has a grid unfolding: its surface may be unfolded to a single non-overlapping piece by cutting along grid edges defined by coordinate planes through every vertex

O\u27Rourke, Joseph

Smith College: Smith ScholarWorks

Smith ScholarWorksComputer Science: Faculty Publications Computer Science7-12-2007Unfolding Orthogonal TerrainsJoseph O'RourkeSmith College, jorourke@smith.eduFollow this and additional works at: https://scholarworks.smith.edu/csc_facpubsThis Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. Formore information, please contact scholarworks@smith.eduRecommended CitationO'Rourke, Joseph, "Unfolding Orthogonal Terrains" (2007). Computer Science: Faculty Publications. 52.https://scholarworks.smith.edu/csc_facpubs/52Unfolding Orthogonal TerrainsJoseph O’Rourke∗June 18, 2013AbstractIt is shown that every orthogonal terrain, i.e., an orthogonal (right-angled) polyhedron based on a rectangle that meets every vertical line ina segment, has a grid unfolding: its surface may be unfolded to a singlenon-overlapping piece by cutting along grid edges defined by coordinateplanes through every vertex.1 IntroductionThis paper is concerned with unfolding the surface of a polyhedron to a single,connected planar piece that avoids overlap. We will concentrate on orthogonalpolyhedra: those whose faces meet at angles that are multiples of 90◦, and whoseedges are parallel to Cartesian xyz-axes. Figure 1 shows an edge unfolding ofan orthogonal polyhedron, an unfolding produced by cutting along edges of thepolyhedron. Note that we permit boundary overlap, but no interior points ofthe planar piece overlap. Thus the shape could be cut out of paper and foldedup to form the surface of the polyhedron.The study of unfolding orthogonal polyhedra was initiated in [BDD+98], andthere are now many results, which we will not survey (see [O’R07] and [DO07]).It will suffice here to note that an easy example (a small box in the center of alarger box’s top face) demonstrates that not every orthogonal polyhedron maybe edge-unfolded. Consequently, loosenings of the unfolding criteria have beenexplored. A grid unfolding adds edges (grid edges) to the surface by intersectingthe polyhedron with planes parallel to Cartesian coordinate planes throughevery vertex, as in Figure 1(c), permitting cutting along these grid edges. Eventhis freedom has not proven sufficient to obtain broadly applicable algorithms,so grid refinements have been studied. A k1×k2 refinement of a surface [DO05]partitions each face into a k1×k2 grid of faces (with the convention that a 1×1refinement is an unrefined grid unfolding). Athough there have now been severalgrid refinement algorithms developed that unfold special classes of orthogonalpolyhedra (surveyed in [O’R07]), it remains unknown whether every orthogonal∗Dept. Comput. Sci., Smith College, Northampton, MA 01063, USA. orourke@cs.smith.edu.1arXiv:0707.0610v4  [cs.CG]  12 Jul 2007(a)fronttoprightabcd(c)left fronttoprightleft bottom(b)a b cdbackFigure 1: (a) An orthogonal polyhedron. (b) An edge unfolding of the polyhe-dron in (a). (c) Grid edges added to (a) by intersecting with coordinate planesthrough every vertex.polyhedron has a (1×1) grid unfolding. This paper shows that a special classof orthogonal polyhedra does have a grid unfolding.This class we call orthogonal terrains. Let P be the surface of an orthogonalpolyhedron, and P the closed, solid whose boundary is P . An orthogonal terrainsatisfies two properties: (1) there is a distinguished rectangular face of P calledthe base B; and (2) every vertical line L (parallel to the z-axis) that intersects Pmeets it in a single segment, L∩P = s, with s a finite-length line segment withone endpoint on B: s ∈ B. P \B is a “monotone surface,” and models a terrainof elevations. In fact, any Digital Elevation Model (DEM), i.e., any rectangulararray of heights, can be viewed as an orthogonal terrain (when closed with sidesand a base). Figure 2 shows an example we will use throughout the paper(Figure 1(a) is not a terrain because its base is not a rectangle).A slightly broader class of shapes, the “Manahattan towers,” were studiedin [DFO05]. These differ from terrains only in permitting the base B to be anarbitrary orthogonal polygon. This apparently small generalization considerablycomplicates the situation, and that paper achieved only a 5×4 grid unfolding.Insisting that B be a rectangle permits a completely different, and relativelysimple algorithm to achieve a 1×1 grid unfolding.2 Terrain Unfolding AlgorithmWe now proceed to describe that algorithm, relying on illustrations to avoidexcessive formality. Unlike most unfolding algorithms, this one can be specifiedas a continuous motion that avoids self-intersection throughout (as opposed toonly guaranteeing nonoverlap at the planar conclusion). The first two steps2xyz10321032nnFigure 2: A orthogonal terrain with grid edges added, in this case via a planeat every integer coordinate. The base B underneath is a 10×10 square.are straightforward. First, the right (+x), left (−x), and back (+y) verticalfaces are unfolded to the xy-plane while remaining attached to the base B. SeeFigure 3. Second, B and its attachments are rotated around the x-axis, andFigure 3: Unfolding the right, left, and back sides of P .then the front vertical faces unfolded horizontally as in Figure 4. Here the lineof rotation is x = 0∩ z = h, where h is the height of the tallest front face (h = 3in the figure; six front faces are tied for tallest).All this is straightforward. The third step of the algorithmm is the heartof it. Define an xi-strip as the sequence of faces between y = i and y = i + 1(i = 0, . . . , n−1) on the “top” of P : the horizontal xy-faces, and the verticalyz right and left faces connecting them in a sinuous path. Each xi-strip willbe unfolded as a unit, into a (long) rectangle stretching in the x-direction. Forexample, the first x0-strip (covering y = [0, 1]) in Figure 2 unfolds to a 16×1rectangle: n = 10 unit square top xy-faces, connected by right/left pairs of 1×2and 1×1 vertical faces. See Figure 6.Consider any adjacent x-strips, xi−1 and xi. In the original P , they areconnected by a number of vertical xz-faces, some rising at y=i to connect to3Figure 4: Flipping the base B around the line y=z=0, and then unfolding thefront faces of P .Figure 5: Unfolding the top faces of P into x-strips connected by y-bridges.4a higher y-adjacent “tower,” and some descending to connect to a lower y-adjacent neighbor. Define the bridge bi to be the xz-rectangle of greatest z-height between the strips, breaking ties arbitrarily. Then we lay out the xi-stripseparated from the xi−1 strip by the height of bi in the planar vertical (y-)direction, and aligned horizontally so that bi connects the two strips. Note thatall the connecting xz-rectangles are attached above the xi strip. The continuousunfolding process is depicted in Figure 5, and the final unfolding is shown inFigure 6. Note that, because of ties, the unfolding is not a simple polygon;xyFigure 6: The final unfolding of P from Fig. 2 in the xy-plane.rather, the boundary overlaps at several places. However, the unfolding is whatis known as weakly simple, in that no interior points overlap, as mentionedpreviously.53 ConclusionAlthough our example gridded the polyhedron at every integer lattice point,it is clear that a coordinate grid plane through every vertex suffices for thealgorithm.Orthogonal terrains add to the narrow classes of orthogonal polyhedra thatare known to be grid-unfoldable (orthotubes, well-separated orthotrees, orthog-onally convex orthostacks; see [O’R07]), although it may be that all orthogonalpolyhedra may be grid-unfoldable. Even extending this new algorithm to ter-rains defined by slanted axes (e.g., Figure 7) remains problematical.xy(a) (b)x'z'y'yz30°z'...012012Figure 7: (a) The polyhedron from Fig. 2 with the z axis slanted 30◦ towardthe y-axis. (b) Partial unfolding of first three strips {x0, x1, x2}, showing thatthe algorithm that produced Fig. 6 now leads to overlap.Acknowledgments. I am indebted to Mirela Damian and Robin Flatland,whose work in [DFO05] led to the algorithm in this paper. I thank StefanLangerman for the slanted axes question.References[BDD+98] Therese Biedl, Erik D. Demaine, Martin L. Demaine, Anna Lubiw,Joseph O’Rourke, Mark Overmars, Steve Robbins, and Sue White-sides. Unfolding some classes of orthogonal polyhedra. In Proc.10th Canad. Conf. Comput. Geom., pages 70–71, 1998. Full versionin Elec. Proc.: http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-biedl-unfolding.ps.gz.[DFO05] Mirela Damian, Robin Flatland, and Joseph O’Rourke. UnfoldingManhattan towers. In Proc. 17th Canad. Conf. Comput. Geom.,pages 204–207, 2005. Full version: arXiv:0705.1541v1 [cs.CG].6[DO05] Erik D. Demaine and Joseph O’Rourke. Open problems from CCCG2004. In Proc. 17th Canad. Conf. Comput. Geom., pages 303–306,2005.[DO07] Erik D. Demaine and Joseph O’Rourke. Geometric Folding Algo-rithms: Linkages, Origami, Polyhedra. Cambridge University Press,June 2007. http://www.gfalop.org.[O’R07] Joseph O’Rourke. Unfolding orthogonal polyhedra. In J.E. Good-man, J. Pach, and R. Pollack, editors, Proc. Snowbird ConferenceDiscrete and Computational Geometry: Twenty Years Later. Amer-ican Mathematical Society, 2007. To appear.7

Unfolding Orthogonal Terrains

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Unfolding Orthogonal Terrains

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